Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340

Warning: preg_replace(): Compilation failed: invalid range in character class at offset 4 in /home/hitowomusubu/www/wp/wp-content/plugins/crayon-syntax-highlighter/crayon_langs.class.php on line 340
三角形の角の問題の解法 | 夢を叶える塾
スポンサーリンク

三角形の角の問題の解法

ホームページのリニューアルに伴い一部のページは新しいサイトにリダイレクトされます。

 

三角形の内角の和と外角の和の性質を使った問題です

基本問題

(1)次の図の\angle{CBD}の値を求めなさい。
三角形の角_1
\angle{BAC}=60^{\circ}
\angle{BCA}=50^{\circ}

(2)次の図の\angle{CDA}の値を求めなさい。
三角形の角_2
\angle{BAE}=30^{\circ}
\angle{EBA}=70^{\circ}
\angle{ECD}=35^{\circ}

(3)次の図の\angle{ECF}の値を求めなさい。
三角形の角_3
\angle{ABE}=80^{\circ}
\angle{BAE}=40^{\circ}
\angle{DFE}=140^{\circ}

解き方

図形の性質を使ってわかる角や線の関係や値を埋めていきます。

解説

(1)次の図の\angle{CBD}の値を求めなさい。
二つの内角の和は残る角の外角に等しい性質を使います。
\angle{CBD}=\angle{BAC}+\angle{BCA}にそれぞれ代入します。
\angle{CBD}=60^{\circ}+50^{\circ}=110^{\circ}

(2)次の図の\angle{CDE}の値を求めなさい。
三角形の内角の和が180^{\circ}となる性質\triangle{BAE}に使います。
\angle{BAE}+\angle{EBA}+\angle{BEA}=180^{\circ}にそれぞれ代入します。
30^{\circ}+70^{\circ}+\angle{BEA}=180^{\circ}
100^{\circ}+\angle{BEA}=180^{\circ}
\angle{BEA}=80^{\circ}

対頂角が等しくなる性質を使います。
\angle{BEA}=\angle{CED}なので\angle{CED}=80^{\circ}

三角形の内角の和が180^{\circ}となる性質\triangle{CED}に使います。
\angle{CED}+\angle{ECD}+\angle{CDE}=180^{\circ}にそれぞれ代入します。
80^{\circ}+35^{\circ}+\angle{CDE}=180^{\circ}
115^{\circ}+\angle{CDE}=180^{\circ}
\angle{CDE}=65^{\circ}

(3)次の図の\angle{ECF}の値を求めなさい。
二つの内角の和は残る角の外角に等しい性質\triangle{BEA}に使います。
\angle{FEC}=\angle{BAE}+\angle{ABE}にそれぞれ代入します。
\angle{FEC}=40^{\circ}+80^{\circ}=120^{\circ}

合成した角が直線となるのは合成した角が180^{\circ}となる性質\triangle{DFE},\triangle{EFC}に使います。
\triangle{DFE}+\triangle{EFC}=180^{\circ}\angle{DFE}=140^{\circ}を代入します。
140^{\circ}+\triangle{EFC}=180^{\circ}
\angle{EFC}=40^{\circ}になります。

三角形の内角の和が180^{\circ}となる性質\triangle{CEF}に使います。
\angle{CEF}+\angle{EFC}+\angle{FEC}=180^{\circ}にそれぞれ代入します。
40^{\circ}+\angle{EFC}+120^{\circ}=180^{\circ}
160^{\circ}+\angle{EFC}=180^{\circ}
\angle{CDE}=20^{\circ}

応用問題

(1)次の図の\angle{CBA}の値を求めなさい。
三角形の角_4

\angle{BAD}=20^{\circ}
\angle{ADC}=40^{\circ}
\angle{DCB}=10^{\circ}

(2)次の図の\angle{ABD}の値を求めなさい。
三角形の角_6

\angle{BAD}=30^{\circ}
BA=BC
AD=AC

解き方

補助線をひくことで答えを求めやすくなることがあります。
二等辺三角形、正三角形のときは辺の長さから角の情報を得ることができます。
基本問題同様、図形の性質を使ってわかる角や線の関係や値を埋めていきます。

埋めることができなくなった場合、文字を使って方程式を使って解いていきます。

解説

(1)次の図の\angle{CBA}の値を求めなさい。
楔形のときは、補助線を引くと求めやすくなることがあります。
BDに補助線を引きましょう。

三角形の角_7

これで三角形ができます。

二つの内角の和は残る角の外角に等しい性質\triangle{BDC}に使います。
\angle{EBC}=\angle{BDC}+\angle{BCD}に代入します。
\angle{EBC}=\angle{BDC}+10^{\circ}
\angle{BDC}が残っていますが、これは\angle{BDA}とくっつけば40^{\circ}になります。
この中途半端に見える式も重要な情報になります
だから何でも手を動かしてほしいと思っているんですね

二つの内角の和は残る角の外角に等しい性質\triangle{BDA}に使います。
\angle{EBA}=\angle{BDA}+\angle{BAD}に代入します。
\angle{EBC}=\angle{BDA}+20^{\circ}
\angle{BDA}が残っていますが、これは\angle{BDC}とくっつけば40^{\circ}になります。
という事でくっつきそうですね。

\angle{CBA}=\angle{EBC}+\angle{EBA}=\angle{BDC}+10^{\circ}+\angle{BDA}+20^{\circ}
=\angle{BDC}+\angle{BDA}+30^{\circ}
という事で\angle{BDC},\angle{BDA}がくっつきます。
=40^{\circ}+30^{\circ}
=70^{\circ}

もう一つ、三角形と思わず、平行線を使っても求める事ができます。
ADに平行な直線でBCを通る直線を引きます。
三角形の角_8

平行線の錯角は等しい性質より\angle{BAD}=\angle{EBA}=20^{\circ}になります。
また同じく、\angle{ADC}=\angle{FCD}=40^{\circ}になります。
さらに、\angle{CBE}=\angle{FCB}になります。

\angle{FCB}=\angle{FCD}+\angle{DCB}なので、これに代入します。
\angle{FCB}=40^{\circ}+10^{\circ}=50^{\circ}になります。

\angle{CBE}=\angle{FCB}だったので、\angle{CBE}=50^{\circ}です。

\angle{CBA}=\angle{EBC}+\angle{EBA}=50^{\circ}+20^{\circ}=70^{\circ}

(2)次の図の\angle{ABD}の値を求めなさい。
\triangle{ABC},\triangle{ACD}BA=BC,AD=ACより二等辺三角形になっています。
二等辺三角形の二つの底角は等しい性質より\angle{BAC}=\angle{BCA},\angle{ACD}=\angle{ADC}になります。
これより\angle{BAC}=\angle{ADC}になります。

これ以上埋まるところが無いですね。
x=\angle{ABC}とします。

二つの内角の和は残る角の外角に等しい性質\triangle{BDA}に使います。
\angle{CDA}=\angle{DBA}+\angle{DAB}になります。
\angle{CDA}=x+30^{\circ}
\angle{BCA}=\angle{CAB}=\angle{CDA}=x+30^{\circ}であることがわかりました。

三角形の内角の和が180^{\circ}となる性質\triangle{ABC}に使います。
\angle{ABC}+\angle{BCA}+\angle{CAB}=180^{\circ}です。
x+(x+30^{\circ})+(x+30^{\circ})=180^{\circ}
3x+60^{\circ}=180^{\circ}
3x=120^{\circ}-x
x=40^{\circ}
\angle{ABC}=40^{\circ}になります。

終わりに

わかりそうなところから埋めていけば大体答えにたどり着けます。
埋まらないときは平行線を引いてみたり、対角線を引いてみたりしてみましょう。
もしくは方程式を作って解いてみましょう。
特に二等辺三角形や正三角形等の「等しい角」というのは方程式を使う場面が多くなります。

にほんブログ村 地域生活(街) 東北ブログへ にほんブログ村 地域生活(街) 東北ブログ 米沢情報へ にほんブログ村 教育ブログへ にほんブログ村 教育ブログ 塾教育へ
スポンサーリンク